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A perfect square is a number or expression that can be expressed as the product of an integer or algebraic term with itself. For example, some numbers like 4, 9, and 25 are perfect squares as they can be written as \(2^2\), \(3^2\), and \(5^2\) respectively. Recognizing perfect squares is crucial when simplifying square roots, because the square root of a perfect square is always an integer.

In our exercise, 25 is a perfect square as it can be represented by \(5^2\). Similarly, for algebraic expressions, \((a+4)^2\) is a perfect square since it's the product of \((a+4)\) with itself. Knowing that these components within the square root are perfect squares allows us to take them out of the root during simplification, making the process much more straightforward.

Understanding square root properties allows for efficient simplification of radicals. One key property used in our exercise is that the square root of a product can be written as the product of the square roots: \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\). By applying this property, you can simplify square roots by separating them into simpler parts.

However, it is important to mention that this only holds true when the numbers or expressions inside the square root are non-negative since the square root of a negative is not a real number. In applying this to the exercise, we split the square root of the product of 25 and \((a+4)^2\) into the product of the separate square roots of 25 and \((a+4)^2\), enabling us to simplify each square root individually.

The absolute value of a number or expression represents its distance from zero on a number line, without considering the direction. It is always a non-negative value. The symbol for absolute value is a pair of vertical bars: \(|a|\). It is particularly relevant when dealing with square roots of squares, as the square root of a squared term yields the absolute value of the term.

This is due to the fact that squaring a number and then taking the square root should ideally return the original number. However, since we cannot have a negative result from a square root, we return the absolute value to ensure it is non-negative. In our provided exercise, after applying the square root to \((a+4)^2\), we get the absolute value of \(a+4\), as we cannot be certain of the sign of the term without further information about the variable \(a\).